Mathematische Annalen

, Volume 374, Issue 1–2, pp 653–680 | Cite as

On the Fourier spectrum of functions on Boolean cubes

  • Andreas Defant
  • Mieczysław Mastyło
  • Antonio PérezEmail author


Let f be a real-valued function of degree d defined on the n-dimensional Boolean cube \(\{ \pm 1\}^{n}\), and \(f(x) = \sum _{S \subset \{1,\ldots ,n\}} \widehat{f}(S) \prod _{k \in S} x_k\) its Fourier-Walsh expansion. The main result states that there is an absolute constant \(C >0\) such that the \(\ell _{2d/(d+1)}\)-sum of the Fourier coefficients of \(f:\{ \pm 1\}^{n} \longrightarrow [-1,1]\) is bounded by \(C^{\sqrt{d \log d}}\). It was recently proved that a similar result holds for complex-valued polynomials on the n-dimensional polytorus \(\mathbb {T}^n\), but that in contrast to this, a replacement of the n-dimensional torus \(\mathbb {T}^n\) by the n-dimensional cube \([-1, 1]^n\) leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.


Boolean functions polynomial inequalities Fourier analysis on groups 

Mathematics Subject Classification

06E30 47A30 81P45 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikCarl von Ossietzky UniversitätOldenburgGermany
  2. 2.Faculty of Mathematics and Computer SciencesAdam Mickiewicz University in PoznańPoznańPoland
  3. 3.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus Cantoblanco UAMMadridSpain

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